1. What hypothesis was tested?
The null hypothesis in this situation was that the distribution of the signs of the zodiac would essentially be equal because people reproduce at a constant rate year round. Therefore the alternative hypothesis was that the students would fall under one or a couple zodiac signs because humans do not reproduce at the same rate year round. Humans would be more likely to reproduce during the spring months because in nature most animals are born during that time period because there is an abundance of food during this period, so it would be best to raise their young in. If this hypothesis were true, then people would be more likely to be born from February to June, which ranges across the zodiac signs of Pisces, Aries, Taurus, and Gemini. The hypothesis tested in this situation was the null hypothesis. Scientists can never prove that the null hypothesis is correct; they can only disprove it and therefore accept the alternative. In other words, one can never prove the alternative hypothesis true, only the null hypothesis false.
2. Did the observations support the hypothesis?
The observations, after analysis, failed to refute the null hypothesis. When looking at the raw data, though, it appeared that because both Pisces and Aries contained eight people in their respective groups, which was more than the expected outcome. However, when averaging together the number of people in Pisces, Aries, Taurus, and Gemini, it was found that there were, on average, 6.25 people born under each zodiac sign in the spring. This was less than the expected number of people for each zodiac sign, which was set at 7.17 people. The critical probability value, taking into account the fact that there were eleven degrees of freedom and that the null hypothesis would be rejected at 95% certainty, was set at 19.675. When the sum of the chi square values was found, it equaled 15.302, which was less than the critical probability value. Thus, the null hypothesis was supported.
3. What observations were made (data and data analysis)?
In this lab, the zodiac signs for a sample size of 86 students were observed. This data was then organized into groups, like signs grouped together. The expected number of people for each zodiac group was also observed. In this case, because there was a sample size of 86 people and twelve zodiac signs, it was expected that there would be 7.17 people in each group. Of course, this number would be impossible to attain because a person can’t be split into parts, but as long as the chi square value was less than the critical probability value, then the observations would support the null hypothesis in that situation. In fact, all of the observations supported the null hypothesis, that the number of people in the sample would essentially be distributed evenly in each zodiac sign. This was determined by subtracting the expected value, which was attained through mathematical calculation, from the observed value, which was attained through the survey. The subsequent number was then squared and divided by the expected value to come up with the chi square value. The sum of chi square values were then found, which was 15.302. The critical probability value was calculated to be 19.675 using the chart. There were eleven degrees of freedom in this lab because there were twelve different outcomes, and one was subtracted from twelve to determine the degrees of freedom. The level of significance was set to be 5% or 0.05, meaning that the null hypothesis would be accepted until one was certain that it was 95% wrong. The value in this location of the table provided was 19.675. Overall, the null hypothesis was supported.
At first glance, however, there was some deviation within the zodiac signs compared to the expected outcome (and compared to the alternative hypothesis, for that matter, because none of the major deviations where there was a surplus of people occurred in the spring months, and there was even a deviation to the low end of the spectrum in the spring months, which was Aries), which was 7.17 people in each zodiac sign (if the signs were evenly distributed over the sample size of 86 people). The zodiac sign with the largest individual chi square value was Aquarius, which, with 15 out of the sample of 86, had a chi square value of 8.55. This significant deviation from the expected number of people in each zodiac group accounted for about 56% of the total chi square value. The next largest chi square value was Aries, which only contained three reported members of the group and had a chi square value of 2.43. This accounted for about 16% of the total chi square value. The smallest chi square value was Sagittarius, which consisted of seven people out of the entire sample (only 0.17 off from the expected number of people in each zodiac sign, which was 7.17 people) and had an individual chi square value of 0.004.

4. What conclusion was made from the analysis?
The conclusion drawn from the analysis was that the null hypothesis, that the amount of people with each zodiac sign would be evenly distributed because people reproduce at the same rate year round, was supported. The alternative hypothesis, that students would be more likely to be born under one or a few zodiac signs because of birthing patterns perpetuated by nature (thus it would be more likely for people to be born in spring), was not supported. Thus, the “status quo” of the null hypothesis was maintained. The sum of the chi square values in this instance was 15.302, less than the critical probability value, which was calculated to be 19.675. In fact, nine of the twelve zodiac signs contained either 5, 6, 7, or 8 people in them, which was close to the expected number of people in each group, 7.17 people. The few significant deviations from the expected value, Aquarius with 15 people, Leo with 11 people, and Aries with 3 people, can be accounted for by the fact that the sample size was 86 people. If the sample size had been larger, then it would have been more likely for the students to be evenly distributed across the zodiac signs, and one would have more confidence in their results. This is because the sample size would have been more likely to include more evenly distributed demographic groups, and as one increases the number of people, the sample size become a larger percentage of the population. If the sample size is a larger percentage of the population, then it is more likely to reflect the trends in the population. Because the null hypothesis was that humans reproduce at the same rate every year, it can be assumed that the population would reflect this because the null hypothesis represents the status quo, or the general consensus on what is accepted.
5. How would increasing the sample size (total number of observations) affect the confidence one has in the conclusion from the analysis?
Increasing the sample size would increase the confidence in the analysis. This is because the larger the sample, the more likely it is to reflect the composition of the population. If the sample size is smaller, it is more likely that individual quirks will affect the data. When the sample size is increased, it is more likely that certain demographic groups will not influence the data as much, that one is drawing upon from all of the demographics. This can also be proven mathematically using the calculation for standard error.

This equation represents standard error, or the amount of deviation from the results acquired through sampling compared to population. S represents the population and accuracy (or standard deviation from that of the population) of the proportion or statistic while N represents the sample size. As N, or the sample size, increases, one can see that the divisor of the population will decrease, and thus the quotient of the standard error equation. If N were to decrease, then the result of the equation would increase, therefore the standard error would increase, and the assumed accuracy of the experiment would decrease.

Works Cited
NCalculators.com. "Calculate Standard Deviation from Standard Error." Finance Calculators. NCalculators.com, 1 Sept. 2012. Web. 11 Sept. 2012. <http://ncalculators.com/math-worksheets/calculate-standard-deviation-standard-error.htm>.